Problem

(a) Show that

$$\left(\cosh x+\sinh x\right)^{\frac{1}{2}}=e^{\frac{1}{2}x}.$$

(b) Find the particular solution of the differential equation

$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+\frac{\mathrm{d}y}{\mathrm{d}x}+3y=5\left(\cosh x+\sinh x\right)^{\frac{1}{2}},$$

given that, when $x=0$, $y=1$ and $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{4}{3}$.